Symplectic forms in the theory of solitons

Abstract

We develop a Hamiltonian theory for 2D soliton equations. In particular, we identify the spaces of doubly periodic operators on which a full hierarchy of commuting flows can be introduced, and show that these flows are Hamiltonian with respect to a universal symplectic form $ω={1\over 2}\r_{\infty} <Ψ_0^*δL\wedgeδΨ_0>\d k$. We also construct other higher order symplectic forms and compare our formalism with the case of 1D solitons. Restricted to spaces of finite-gap solitons, the universal symplectic form agrees with the symplectic forms which have recently appeared in non-linear WKB theory, topological field theory, and Seiberg-Witten theories. We take the opportunity to survey some developments in these areas where symplectic forms have played a major role.

arXiv comment76 pages, Tex, no figures

References in corpus (9)

• Phases of $N=2$ Theories In Two Dimensions
• Supersymmetric Yang-Mills Systems And Integrable Systems
• The Vacuum Structure of N=2 SuperQCD with Classical Gauge Groups
• Prepotentials of $N=2$ Supersymmetric Gauge Theories and Soliton Equations
• Introduction to Seiberg-Witten Theory and its Stringy Origin
• Whitham-Toda hierarchy and N = 2 supersymmetric Yang-Mills theory
• Integrable Structures in Supersymmetric Gauge and String Theory
• Non-perturbative Quantum Theories and Integrable Equations
• Non-perturbative aspects of (supersymmetric) gauge theories

Cited by in corpus (1)

• On the Dynamics of Finite-Gap Solutions in Classical String Theory